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Best constants for the isoperimetric inequality in quantitative form

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Marco Cicalese, Gian Paolo Leonardi

Analysis of PDEs · Submitted 30 Dec 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals.

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arXiv ID1101.0169 (version 1)
Pith ID257IHHJ7, Pith's permanent record of this paper and its review
CitationGian Paolo Leonardi, Marco Cicalese. "Best constants for the isoperimetric inequality in quantitative form." arXiv:1101.0169 (2010). https://pith.science/paper/257IHHJ7
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,738 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.62 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. result 0.71 0.95 0.13 asserted 1.00
s1 Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. method 0.68 0.77 0.23 asserted 0.95
s2 Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. result 0.47 0.95 0.30 asserted 1.00
s3 This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992). interpretation 0.80 0.80 0.33 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).

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